An integration of ∞-Lie algebroid valued differential forms $\omega : \Pi^{inf}(X) \to \mathfrak{a}$ is an extension $\int \omega$ from the infinitesimal path ∞-groupoid? $\Pi^{inf}(X)$ to the finite path ∞-groupoid $\Pi(X)$
after injecting the ∞-Lie algebroid $\mathfrak{a}$ into one of its corresponding ∞-Lie groupoids $A$.
Assume a context given by a smooth (∞,1)-topos with its notion of ∞-Lie theory.
For $A$ some ∞-Lie groupoid and $U \in C$ a test domain, a morphism $\phi : \Pi^{inf}(U) \to A$ from the infinitesimal path ∞-groupoid? factors – by definition of ∞-Lie algebroid – through the ∞-Lie algebroid $\mathfrak{a}$ of $A$
The first morphism here constitutes the set of ∞-Lie algebroid valued differential forms on $U$ that is encoded by $\phi$.
We want to specify what it means to integrate these forms.
Whereas $\omega$ colors infinitesimal intervals on $U$ by infinitesimal morphisms in $A$, its integration should be a rule that labels finite intervals by finite morphisms, such that the restriction to any infinitesimal interval within a finite interval reproduces the assignment of $\omgea$.
Formally, this desideratum clearly means that integration $\int \omega$ of $\omega$ is an extension
of $\omega$ through the inclusion $\Pi^{inf}(U) \hookrightarrow \Pi(U)$ of the infinitesimal path ∞-groupoid? in the path ∞-groupoid.
More generally, we can ask for a relative integration : we may have a situation where a coloring of finite intervals
is already given, but infinitesimally there is also a lift specified of this through some map $A \to B$, i.e. a diagram
Integration should exist when the map $A \to B$ in principle admits such a lift and should be given by that lift. It should be unique up to equivalence.
We discuss a case in which integration of $\infty$-Lie algebroid valued forms always exsist.
Let $\mathfrak{a}$ be an ∞-Lie algebroid.
Write for short $\mathfrak{a}_# := \mathfrak{a}_{#^{inf}}$ in the following for the image of $\mathfrak{a}$ under the right adjoint of the infinitesimal path ∞-groupoid? functor $\Pi^{inf}$. And write $\mathfrak{a}_{# 0}$ for the corresponding presheaf of 0-cells.
By the reasoning an ∞-Lie differentiation and integration we have
the $\infty$-Lie algebroid of $\mathfrak{a}$ is $\mathfrak{a}$ itself:
the ∞-Lie groupoid integrating $\mathfrak{a}$ is
For $X$ simplicially discrete, every morphism
corresponding by adjunction to a morphism
induces a morphism
that fits into the diagram